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PACIFIC JOURNAL OF MATHEMATICSVol. 91, No. 1, 1980

THE SCHEME OF FINITE DIMENSIONALREPRESENTATIONS OF AN ALGEBRA

KENT MORRISON

For a finitely generated /̂ -algebra A and a finite dimension-al ^-vector space M the representations of A on M form anaffine ^-scheme Mod^(M). Of particular interest for this schemeare the connected components, the irreducible components, andthe open and closed orbits under the natural action of thegeneral linear group AutA(M), since the orbits are the equiva-lence classes of representations. The connected componentsare known for a finite dimensional algebra A. In this paperwe characterize the connected components when A is com-mutative or an enveloping algebra of a Lie algebra in chara-cteristic zero. For the algebra k[x> y]/(x, y)2 we describe theopen orbits and the irreducible components. Finally, we ex-amine the connection with the theory of deformations of al-gebra representations.

Introduction* For almost any ^-algebra A it is an impossibletask to classify all finite dimensional A-modules. There are somestandard exceptions such as the finite dimensional semisimplealgebras, A = k[x]/(xm), A = k[x, y]/(x, yf, and A = k[x]. Even forthe polynomial algebra in two indeterminates, k[x, y], Gelfand andPonomarev have shown that the classification problem is as difficultas classifying the modules over the free algebra k{x19 •••, xm}. (See[7] for details.)

If we restrict our attention to modules of a given dimension,say n, then to classify the ^-dimensional A-modules is to classifythe orbits of Aut}χkn) = GL(n) acting on the space of ^-dimensionalA-module structures. If we take A to be a finitely generated k-algebra then this space of module structures is an affine fc-scheme,which we call Mod^(M) where M = kn. Although we cannot hopeto determine the orbits in most cases, we may be able to describecoarser features of ModA(M) such as the irreducible components andthe connected components. It is surprising that even for the con-nected components there is not a complete answer for an arbitraryfinitely generated ^-algebra, while the irreducible components arenot at all understood. Only for one interesting algebra, namelyA = k[x, y]/(x, y)2, have the irreducible components been determined,and for this it is essential to know the classification of the finitedimensional A-modules. (In § 5 we recall the description of theindecomposable modules from [9], determine all open orbits, andfrom them describe the irreducible components as found by Flanigan

199

200 KENT MORRISON

and Donald [5].)More is known about the connected components of Mod^(ilί).

Assume k is algebraically closed in order to simplify the discussion.If A is finite dimensional over k, then two module structures arein the same connected component if and only if they have isomor-phic Jordan-Holder factors [6]. Therefore the connected componentscorrespond to the semisimple modules. This is an effective criterion,too, because for each simple ^.-module L the multiplicity of L inthe composition series for an A-module structure p is given by therank of p(e) where e is a minimal idempotent such that L = Ae/Neand N is the radical of A. Thus, if p is given as a set of matricessatisfying the relations of A, then one can compute rk p{e) for anorthogonal set of minimal idempotents and thereby determine whentwo module structures are in the same component.

In § 3 and § 4 we also describe the connected components whenA is a finitely generated commutative algebra and when A is theenveloping algebra of a finite dimensional Lie algebra in character-istic zero. These are both infinite dimensional examples, but theydo not point to any obvious conjectures that would generalize theresults.

One may view Mod^(itf) as a generalization of Spec A for non-commutative algebras. In fact, for A commutative and k algebrai-cally closed M.oάA(k) — Spec A, A better generalization would bethe quotient MoάA(M)fAutk(M) but it is not a scheme. There is,however, a categorical quotient X whose points are the isomor-phism classes of semisimple A-modules as shown in the work ofProcesi [14, 4.1], The projection π: Mod (̂Af) —• X maps two modulestructures to the same point if and only if they have isomorphicJordan-Holder factors. Thus we see that when A is finite dimen-sional, X is the space of connected components of Moduli). Forthe special case A = k[x] see Mumford and Suominen [12] andByrnes and Gauger [2].

In § 5 we examine the structure of MoάA(M) for the threedimensional algebra k[x, y]/(x, y)2 and describe the open orbits andirreducible components. In § 6 we translate the language of defor-mation theory of modules in the style of [4 and 8] to the geometricsetting of Mod^(M). Several results from [8] about rigidity areproved by constructing corresponding subschemes of Mod (̂ikf).

Throughout this article we use the language of [3].

1* The scheme ModLt(Λf)* Let k be a commutative ring withunit, A an associative ά-algebra and M a ^-module. The functorMod (̂Af) from the category of commutative fc-algebras to thecategory of sets is defined by

THE SCHEME OF FINITE DIMENSIONAL REPRESENTATIONS 201

ModA(M)(R) = the set of A (x) i?-module structures on M0R.

This is also the set of 12-algebra homomorphisms from A (x) R toEndΛ(Af (x) j?) or the set of representations of the j?-algebra A (x) Ron the i?-module M®R.

THEOREM 1.1. If A is a finitely generated k-algebra and M isa protective k-module of finite type, then Mod (̂ikf) is representableby an ajfine k-scheme.

If in addition A is finitely presented or k is noetherian, thenMoάA(M) is an a fine k-scheine of finite type.

Proof. P i c k a s e t o f g e n e r a t o r s alf -—,an f o r A . L e t / =(fj)jej b e t h e i d e a l o f r e l a t i o n s w i t h g e n e r a t o r s fd. F o r e a c h jeJd e f i n e a m o r p h i s m o f f u n c t o r s

x ••• xEnd,(M) > EndΛ(Λf)

(ul9 , un) i • fά(uu , uJ .

The functor Endk(M) is given by Endk(M)(R) - End^Af (x) R) andis representable [3, II, §1,2.4]. Now the zero subfunctor Z aEnd&(Λf) is closed and afBne, so that fi\Z) is closed and affine.Therefore the functor f\jeJfi\Z) is closed and affine. The i?-pointsof this functor are clearly ^-tuples of i2-module endomorphismswhich define i2-algebra homomorphisms from A (x) R to End^Mx)^)by mapping α̂ (x) 1 to ut.

For the finiteness conditions, if A is finitely presented then wecan take J to be a finite set. If k is noetherian then the coordinatering of Endfc(M), which is S(M* 0 M), is also noetherian and sothe ideal defining MoάA(M) in Endk(M)n is finitely generated. •

With the hypotheses of the theorem A κ> Mod^(M) is a con-travariant functor from the category of finitely generated A -algebrasto the category of (affine) A -schemes. An algebra morphism f:A—>B induces the morphism of functors f*:M.oάB(M)->M.oάA(M) whichis "restriction of scalars".

PROPOSITION 1.2. Under the hypotheses of 1.1 let la A be atwo-sided ideal. Then Mod^/7(ilί) c Mod^(M) is a closed siώscheme.

Proof. Let {e0}j£j be a set of generators for /. For each jeJdefine Zά to be t h e closed subscheme whose iϋ-points are t h e A®R-module s t ruc tures p such t h a t ρ(e5®lR) = 0 in End/ 2(M(x) R). Then

Let Antk(M)(R) = {φ: M® R-> M<g> R\φ is an i?-module auto-

202 KENT MORRISON

morphism}. There is a natural action of the &-group scheme Autfc(ilί")on Mod (̂ikΓ) defined for each R by

7Λ: Autk(M)(R) x MoάA(M)(R) • ModA(M)(R)

φ,p\ >φ-p

where (φ pXa) — φ<=>ρ{a)oφ-1 for αeA(x)i?. Therefore φ p and pare isomorphic A (x) ϋJ-modules.

2* Topological properties* We assume that & is a field sothat the orbits of y are subschemes. Let p be a point inModA(M)(k) and let Xp be its orbit. Recall that as a functor Xp isthe sheafification of the functor which assigns to R the set of i ®iZ-module structures on M0R isomorphic to pE9 where pR is thenatural extension of p. [3, III, § 3, 5.4]

THEOREM 2.1. The tangent space at p of ModA{M) is the vectorspace of linear maps v: A-^Endk(M) satisfying v(aλa2) = p(a1)v(a2) +v(a1)p(a2) for all al9 α2 in A. This is the space of one-cocycles of Awith values in the two-sided A-module Endfc(M) under the structuremap p.

Proof. Let k[ε] = k[x]/(x2) be the algebra of dual numbers. Atangent vector at p is an element p of MoάA(M)(k[ε]) lying over p,i.e., such that ModA(M)(π)(ρ) = p, where π:k[ε]->k is the homo-morphism defined by ττ(ε) = 0. Thus, p has the form p(a + εb) =ρ(a) + ε(p(b) + v(a)) for some ^-linear map v: A -> Endfc(ilί). Directcalculation shows that p is a homomorphism iff v satisfies the indi-cated cocycles condition.

THEOREM 2.2. The tangent space at p of Xp is the vectorspace of linear maps v: A —» Endfc(M) such that for some /eEnd^M)v(ά) — [/, P(a)] This is the space of one-coboundaries of A withvalues in Endk(M) under the structure map p.

Proof The tangent space at / of Autk(M) can be identifiedwith End^ilf) so that feEτ\dk(M) determines the &[ε]-point I + εfwith (I + εfXx + εy) = x + ε(f(x) + y). Then yp(I + εf)(a + εb) =(I + εf)op(a + εb)o(I + εf)-\ Using (I + εf)-1 = I - εf, we see that

ΎP(I + ef)(a + εb) = p(a) + ε(p(b) + [/, p(a)]) . Q

The last two theorems show that the normal space at p of Xp

is isomorphic to H\A, Endfc(ikf), p), the first Hochschild cohomologygroup of A with values in Endk(M) via the structure map p. Let


Mp denote M with the A-module structure p. We have the follow-ing isomorphism:

PROPOSITION 2.3. H\A, EndΛ(M), p) ~ Extι

A(Mp, Mp).

Proof. Let v: A —> Endfc(Λf) be a one-cocycle for p and definean extension

0 > M-^~> M® k[ε] -?-> M > 0

i(x) — ex, p(χ + εy) = a;, and A acts on ilf (x) k[ε] by α(# + εi/) =p(a)(x) + ε[p(α)(αj) + ^(α)^)]. It is easy to check that a coboundarydetermines a trivial extension.

In the other direction, given an extension

0 >M-^-*W-?-»M >Q

we pick a splitting over k of W as M φ ε M , so that i(x) = εx,p(x + εy) = x. In order for these to be A-module homomorphismsA must act on MφεM with aeA determining an endomorphismof the form x + εy i—»ρ(a)(x) + ε(v(α)(V) + p(a)(y)) and v must be acocycle. Trivial extensions give rise to coboundaries.

THEOREM 2.4. TΛ,β oriπί X̂ , of an A-module structure p is anopen subscheme of MoάA(M) if and only if H\A, End/;(Af), p) = 0.

Proof. For k algebraically closed it was proved by Gabriel in[6, 1.2]. Therefore assume k is not algebraically closed.

If Xp is open then the tangent spaces of Xp and Mod^(M) areequal at p and so H\A, Enάk(M), p) = 0.

Conversely, assume H\A, Endfc(M), p) = 0. Consider the schemeModj(M) where A = A(g)k9 M — M®k and & is an algebraic closureof k. Now Modi(M) = Mod (̂ikf) x Spec k and the orbit subschemeof pi in ModΛ(M) is X x Spec L The projection

Mod^(M) x Specfe >MoάA(M)

is open and closed so we need to show that Xp x Spec k is open.This follows from the case that the base field is algebraically closedbecause

H\A, End/C(M), pi) = H\A, Enάk(M), p)®k = Q.

THEOREM 2.5. The orbit Xp is a closed subscheme if and onlyif p is a semisimple module.

204 KENT MORRISON

Proof, (i) In [1, 12.6, p. 559] Artin proved that Xp closed im-plies p is semisimple. We give Gabriel's argument [6, 1.3]. Theidea is to deform the module M (under p) to a module isomorphicto soc (M) φ ikf/soc (M), where soc (M) denotes the socle of ikf, whichis the sum of the simple submodules of ikf. Then we continue thisprocess with Af/soc (ikf) and so on, until we arrive at a semisimplemodule, which must happen since the socle is never zero and Mhas finite dimension.

Let V be a complement to S — soc (M) in ikf. Define

9 t : S φ V > S 0 V:(8,v)\ > (ts, v)

and for every ae A, pt(a) = φt°p{a)oφγι. If we let t = 0 then weget a module structure isomorphic to S 0 M/S. This module is inthe closure of the orbit Xp. Therefore if Xp is closed then M =soc (ikf) © ikf/soc (ikf), and so soc (ikf) — M which means M is semi-simple (under p). Continuing the deformation process with M/soc(M)shows that in the closure of Xp are the semisimple modules withthe same Jordan-Holder factors as p.

(ii) For the converse assume p is semisimple. We may con-sider ikf as a module over the finite dimensional algebra B = A/kerp. Since p is semisimple, rad (JB) — 0 and B is a semisimple algebra.All ί?-modules are projective so that Exti(JV, N) = 0 for allj?-modules N. Therefore all orbits of ModB(M) are open in ModB(M).It follows that each orbit must be closed, since each is the comple-ment of the others. By Proposition 1.2 Moduli) is a closed sub-scheme of Mod^M), and we have just shown that Xp is closed inMoάB(M). Therefore Xp is closed in Mod^(lf).

THEOREM 2.6. // A is finite dimensional and A/rad (A) isseparable, then two module structures are in the same connectedcomponent if and only if they have isomorphic Jordan-Holderfactors.

Proof. Let N = raά(A). By the Wedderburn-Malcev Theoremthere is a subalgebra S of A which is isomorphic to A/N. Usingthe injection i: A/N-* A which maps A/N onto S, we get a schememorphism

i*: ModA(M) > ModA/N(M) .

The morphism i* just considers an A-module as an A/iV-module. Iftwo module structures are in the same component of Mod^(M), thentheir images under i* must be in the same component of M.odΛ/N(M)or, equivalently, they must lie in the same orbit of Mod /̂#(ikf) since


A/N is semisimple and the orbits are the connected components.It is easy to see that i*(p) is a semisimple module with the sameJordan-Holder factors as p, since a composition series for p is alsoa composition series for i*(p).

Conversely, if two modules have the same Jordan-Holder factorsthen the closures of their orbits intersect and so their orbits mustlie in the same connected component. •

We would like to generalize this theorem to infinite dimensionalalgebras but it is not true as it stands. For example, let A = &[$].Then MoάA(M) is isomorphic to Endfc(M), a connected scheme. Thereare, however, many nonisomorphic semisimple &[x]-modules of thesame dimension.

If A is a commutative, finitely generated algebra over analgebraically closed field, then we can describe the connected com-ponents of Mod^(M). The proof depends upon the Nullstellensatz.Recall that a commutative ring R is said to be connected if it hasno idempotents other than 0 and 1. Equivalently, the topologicalspace Spec R is connected.

THEOREM 2.7. If A is a connected, commutative, finitely gener-ated algebra over the algebraically closed field k, then MoάA(M) isconnected.

Proof. Fix a basis e19 , en for M. Define the map

/ : ModA«e i» x x Mod A «0) > ModA(M):

(σlf -- σn)\ ^ Θ Θ ^

whose image is connected since Mod f̂X )̂) F& Spec A. The closureof every orbit Xp meets the image of / because Xp contains amodule structure σ which has a composition series Λf0 c M1 c cMn, Mt — (el9 , e<>, since simple A-modules are all of dimensionone by the Nullstellensatz. Then σ deforms to a semisimple moduleMx © MJM^ 'φMJMn_x compatible with the splitting (e^®-' * θ<O This semisimple module is in the image of / and thereforeMod^(M) is connected. •

Recall that a module E over the product algebra Aι x x A3

splits into submodules Eu , Es such that E = £Ί 0 -@ES andeach Et is an Armodule, and for a = (al9 , as) and x = x1 + +:cs we have ax = a1x1 + + asxs.

THEOREM 2.8. Let A be a finitely generated, commutative k-algebra, k algebraically closed. Let A = Aγ X x As where A% is

206 KENT MORRISON

connected, and let p and a he two A-module structures on M. Thenp and σ are in the same connected component if and only if dimMi — dim Ml where Mt and Ml are the Armodules in the decomposi-tions of p and σ.

Proof Let B = k x x h, s factors. We have the algebramorphism ί: B—* A and the scheme morphism "restriction of scalars"i*: ModA(M) -*ModB(Λf). If p and σ are in the same componentthen i*(p) and i*(σ) are in the same component of ModB(M). SinceB is semisimple i*(p) and i*(σ) are isomorphic J5-modules. Theisomorphism type of a i?-module is determined by the dimensionsof the 8 subspaces that occur in its decomposition. (Some of thesubspaces may be 0.) The decomposition of i*(p) coincides withthat of p and similarly for ΐ*(σ). Therefore dim Mt = dim Ml.

Conversely, if dim Mt — dim Ml for 1 <̂ i <L s, then there is amodule structure τ in the orbit of σ having the same decomposi-tion as p. That is, the underlying subspaces are the same. Thenp and τ are in the image of the scheme map

S: UoάΛl iMi) x x Mod^(ikf8) > Mod^(M)

which forms the direct sum of the modules over the product ofthe algebras. Since each of the schemes ModAi(Mt) is connected bythe previous theorem, the product is connected and the image ofS is connected. Therefore p and τ are in the same connected com-ponent, and so p and σ are in the same component. •

In Theorem 2.9 the assumption of commutativity for A isessential as the next section will show. Enveloping algebras ofLie algebras are connected—they have no nontrivial idempotents.They even have skew fields of fractions. However, there areseveral connected components in M.odϋ{L)(M) unless L is solvable.

Theorems 2.6 and 2.8 both characterize the connected componentsin terms of the components of ModB(M) for a subalgebra B. Inthe finite dimensional case B is a subalgebra isomorphic to A/N. Inthe commutative case B is k x - x k, s factors. In both cases Bis a maximal separable subalgebra, and B is unique up to conjuga-tion by a unit. Can one use this observation to describe the con-nected components of ModA(M) more generally?

In the next section is a description of the connected componentsfor an enveloping algebra, but it does not involve a separable,therefore finite dimensional, subalgebra.

3* Representations of Lie algebras* Enveloping algebras ofLie algebras provide concrete examples of infinite dimensional non-


commutative algebras and a good testing ground to try to describethe connected components of ModA(M) for noncommutative algebras.

Let L be a finite dimensional Lie algebra and U(L) its envelop-ing algebra. U(L) is always infinite dimensional. Modules over Lare exactly the same as modules over Σ7(L), so we define ModL(Λf)to be ModUU)(M).

The cohomology groups of L are defined for each left L-moduleN (whereas the cohomology groups for an associative algebra aredefined for two-sided modules) by

H»(L, N) = Extn

U{L)(k, N)

where k is the trivial L-module, L-k = 0. (See [10, p. 234].)We are only interested in the first cohomology group for which

we have the characterization

H\L, N) = Der (L, iV)/Inder (L, N)

the derivations of L in N modulo the inner derivations.Let p: L —» Emdk(M) be a representation of L. We make

Endfc(M) into a left L-module by defining an = [p(a)9 u]. End4(Λf)is also a two-sided C7(L)-module using the associated algebra homo-morphism β: U(L) —> Endk(M). We also have the isomorphism Hι(L,Endk(M), p) = HXU(L), Endfc(ikf), β). For each derivation v:L->EndΛ(Λf) we define the cocycle v: U(L) —> Endfc(Λf), extending v toproducts of elements from L such as v(ab) = p(a)v(b) + v(a)p(b).Clearly, inner derivations become cocycles. In the other direction,a cocycle for U(L), when restricted to L, is a derivation.

Therefore, just as in the case of modules over an associativealgebra, we conclude that the orbit of a representation p is openif and only if Hι(L, Endk(M), p) = 0, and the orbit is closed if andonly if the representation is semisimple.

Now let k be an algebraically closed field of characteristic 0.The connected components of Moduli) are described by this theorem:

THEOREM 3.1. The connected components correspond to theisomorphism classes of L/rad L-modules on M.

Proof. The radical radL is the largest solvable ideal of L.The quotient is semisimple and by Levies Theorem there is a subal-gebra S isomorphic to L/radL. The injection i: L/radL—>Lmapping onto S gives a scheme morphism i*: ModL(ikf)—>ModL/radL(M).If p and σ are in the same connected component of Mod^Λf) thenthey are in the same component of ModL/radL(M). Since L/rad L issemisimple its first cohomology groups are zero (First Whitehead

208 KENT MORRISON

Lemma), and so all the orbits are open. Therefore all the orbitsare closed and the orbits are the connected components. Thusί*(p) is isomorphic to i*(σ).

Now we will show that any representation p can be deformedto one in which radL acts trivially. Let p be a representation ofL. Consider the representation of radL. Since radL is solvablethere exists an eigenvector xeM for the action of radL. Let Xbe the line generated by x and let U be a complement to X. Also,let R = rad L and S — L/rad L viewed as a subalgebra of L. Foreach t e k, t Φ 0, define the linear automorphism of M

φt: I φ U > X® U: (x, u) 1 > (fix, u) .

L e t p t = φt p . T h e n f o r reR

pt(r): x i > p(r)(x)

u 1 > tπz(p(r)(u)) + πu{ρ{r){u)) .

Now let t = 0. We get a representation for which R acts on Mas on the direct sum X0Λf/X, and for S the representation isisomorphic to p\S since S-orbits are closed. We continue in thisway until we have a representation of L with R acting on thesum of one-dimensional modules Xx 0 0 Xμy while the action ofS is isomorphic to the original p. Gail this new representation σ.

Now construct a deformation of σ defined by

σt(s + r) = σ(s) + tσ(r) .

This is actually a representation for each t e k, since

°&rlf r2]) = 0 = [σt(rd, σt(r2)]

for all rx and r2 in R, because the action of R is diagonalized. Lett = 0 and the result is a representation for which R acts trivially.

Therefore in each connected component of Modχ,(Af) is a uniqueclosed orbit of modules with radL acting trivially. Notice, how-ever, that this orbit is not in the closure of the other orbits.

In order to classify the connected components of Mod̂ (Λf) forinfinite dimensional, noncommutative algebras, it would be usefulto have such a characterization for the enveloping algebras inintrinsic terms, without recourse to the structure of the Liealgebras underlying them.

4. Rational points* The basic fact underlying this section isthe following:

PROPOSITION 4.1. Let p be an A-module structure on M. The


rational points of the orbit Xp are the A-module structures isomor-phic to p.

Proof. Let a be a rational point of Xp. Then there is someΦ e Antk(M)(k) such that φ.p^σ, where k is an algebraic closureof k. Now we can assume that ψ is actually in AutΛ(M)(L) forsome finite extension L. For example, let L be the field generatedby the entries of a matrix representation of ψ. Therefore, theextensions pL and σL are isomorphic. But pL ^ p 0 .© p and σL =o © -@σ, each sum with s factors where s = dimkL. By theKrull-Schmidt Theorem one sees that p ~ σ as A-modules. Π

Typically, when an algebraic group acts on a /̂ -scheme therational points in the orbit of a rational point include many pointsthat are not equivalent by a rational point of the group. Wemake use of 4.1 to describe the connected components of the spaceModΛ(M){k) whose topology is the induced Zariski topology.

THEOREM 4.2. Let k be an infinite field. Suppose A is finitedimensional with a decomposition as a semi-direct sum S φ N ofa semisimple subalgebra S and the radical N. Then two modulestructures p and σ are in the same connected component ofΉίodΛ(M)(k) if and only if they have isomorphic Jordan-Holderfactors.

Proof. The algebra morphism S—>A induces a map ModA(M)(k)—»Mods(M)(fe) If two module structures p and σ are in the samecomponent of MoάA(M)(k) then they are in the same component ofMods(ikf)(fc). Because S is semisimple all orbits of M.ods(M) areopen and so M.ods(M)(k) is the disjoint union of a finite number ofopen sets. Therefore each one is closed as well. Each of theseclosed-open sets is the set of rational points of an orbit and so pand σ have the same Jordan-Holder factors.

For the converse we deform p and σ to semisimple modules ps

and σs which are isomorphic. Since k is infinite, these deforma-tions are given by morphisms from the connected set (Spec &[£])(&) = &into ΉίodA{M){k). Therefore ps is in the same component as p andσs is in the same component as σ. Now ps and σs are in the samecomponent because they are in the image of an orbit map withdomain Autk(M)(k), which is also connected since k is infinite.

PROPOSITION 4.3. Let A be finite dimensional. Then the con-nected components of ΉίodA(M) correspond to the Galois equivalenceclasses of semisimple A 0 k-modules on

210 KENT MORRISON

Proof. The components of MoάA(M) x Spec k correspond to thesemisimple A (x) fc-modules. The projection π: Mod (̂ikf) x Spec k—*M.oάΛ(M) maps connected components onto connected components.Since the fibers of the projection are the orbits of the Galois group,two connected components will be mapped to the same componentif they have any Galois equivalent points. •

Let p be a semisimple A-module structure isomorphic toLTι@ ® L ? , each L* is a simple A-module. Then dim X = dimAutk(M) — dim A u t ^ ) , since AutA(p) is the stabilizer of p. We cancompute this using the dimensions of the Lie algebras, since AutA(p)is smooth by virtue of being an open subscheme of the affine spaceΈndA(p). Therefore dim X — n2 — άimkEndA(p) = n2 — Σ cxm\, whereCi is the dimension of

PROPOSITION 4.4. // A is semisimple and άimkA = dimfeikf = n,then \the orbit of the left regular representation has the maximaldimension among the orbits of Mod (̂ikf), and the dimension is

Proof. Let e4 = dimfc Lif ct = dim .̂ End^^), and let ut be themultiplicity of Lt in the left regular representation for the minimalleft ideal Lt. Therefore ut = ejci since nt is the dimension of Lt

over End^(Lί).In order to maximize the orbit dimension, we need to minimize

Σm\Ci over (mlf , ms) e Ns, with the constraint that Σm^i = n.To do this we can use Lagrange multipliers and actually minimizeover Rs.

Define /, g: Rs -> R by

f(xlt ...,χ,) = Σxlβi

g(xlf -, x8) = Σxtet - n .

Now minimize / on the hyperplane given by g(x) = 0. We have

D f ( x l f . . . , x 8 ) = ( 2 c ^ , . . . , 2 c s x s )

D g ( x l f . .-, xs) = (el9 ••-, e.) .

Solving \Df(x) = Dg(x) for λ e R and cc e R% gives 2λcixi = β<β

Therefore xt = ei/2λcέ. Substituting these values into # gives λ =1/2, from which it follows that α?< = β̂ /c* = %<.

The point u = (^, ••-,%.) is actually a minimum since D2f(u)is positive definite and so it is positive definite when restricted tothe tangent space of g~\0) at u. The dimension of this orbit is

u^ = n2 —


5* An example* The structure of Mod4(M) has only beendescribed for a few algebras. One would like to know the irreduci-ble components, the open orbits, and the ordering of orbits givenby Xσ <L Xp if and only if Xo is contained in the closure of Xp.

The details for A = k[x]/(xn) have been worked out by Gabriel[6] in the language of schemes and by Flanigan and Donald [4] inthe language of deformation theory.

We will examine the structure of Mod4(Af) for the threedimensional algebra A = k[x, y]/(x, yf. There is only one connectedcomponent because there is only one semisimple module in eachdimension. In order to determine the open orbits we need to listthe indecomposable A-modules [9, Prop. 5]. In A let uλ — x andu2 ~ y. The radical of A is the ideal U generated by uγ and u2.Let M be an A-module, so UM is a submodule of M. Pick a com-plementary subspace F and write M= VφUM. The scalarmultiplication of uL and u, take the form

0 O'j

L2\ Oj

with respect to the decomposition F φ UM. Any A-module isequivalent to a pair of linear maps in Hom/C(F, W). The moduleis F φ W as a vector space and multiplication by ut has the formof the matrix above with W ~ UM.

Two modules given by the pairs (2\, T.2) and (S19 S2) are isomor-phic if there are automorphisms PeAutk(V) and Q e Autk(W) suchthat S< = QTXP for i = 1, 2.

The indecomposable A-modules fall into three series: En, n ^ 0;Fn, n ^ 0; Gn(π), n ^ 1 and π = ^ or a monic irreducible polynomialin k[x\. These modules are all distinct except that Eo = FQ, the one-dimensional module isomorphic to AjU. The dimensions of theindecomposable modules are

dim En = dim Fn = 2n + 1

dim Gn(π) - 2n(άeg (ττ))f deg (oo) = 1 .

The free module A is isomorphic to F}. The modules En and Fn

are dual.On En the action of u1 and u2 are given by matrices T1 and T«

with F = k%+1 and W = kn.

01 Γ0

212 KENT MORRISON

On Fn we have the transposed matrices:

O 0

"0 . . . 0

L

For the module Gn(π) with π(x) = xm — a^-^™'1 — . . . — a0 thematrices 2\ and T2 are nm x wm square matrices with the follow-ing form

rn r•* 1 J-nn

Ό O O a0

1 O. .O ax

B= 0 l . O α,

LO O-. l α m _ J

Note that B is the companion matrix of π. Now for π = oo we have

"0 . . . 0"

-L 2 *Λ

T2 =

BN

0

_0

-o0

0

0B

N

0

O

o

o

oo

B • •

• NB

• 1 -

• 0

• 0-

T —

0

With this knowledge of the indecomposable A-modules we canstate the theorem describing those modules whose orbits are open.

THEOREM 5.1. Let M he a finite dimensional A-module ofdimension m. The orbit of M is open in Mod^(fcw) if and only ifM is isomorphic to 23£©J57£ + 1 or to Fζ 0 Fi+19 where n ^ l andm = r(2n + 1) + s(2n + 3).

Proof. M is isomorphic to the direct sum of indecomposablemodules I i φ -0/y. Since Ext is additive in both factors

ExtUM, M) = Θ Exti(I;, I,) .

So one needs to determine the Ext-groups for the indecomposableA-modules. The dimension of Exti(J, J) can be computed for inde-composable modules I and J. The computations may be found in[11]. We summarize with a table of the dimensions for Ext^J, J)and we use d(π) for the degree of π.


J

En

Fn

Gκ{φ)

m -\

0

n —

E,

- 2(n = 0)

vi — l ( m

m + n +

nd(φ)

I m -f

+ 1<

2

- i)

< n)

n +0

m —

2(m

(1 =

• n —

Fm

0

= 0)

VII

VII

l(n +

0

n + ΐ)1 m)

min(m,

0

md(τr)

n)d{π)(π — φ)

0 (7r=£0)

From the table one sees that in order for Ext ( 7 0 J, 7 0 J) =Ext (7, 7) 0 Ext (J, 7) 0 Ext (J, 7) 0 Ext (/, J) to be zero, 7 and Jmust be En and 2£Λ+1 or they must be Fn and JPΛ + 1. The theoremthen follows directly with repeated use of this observation. •

In Mod̂ (/bm) for m odd, the irreducible components each havean open orbit and so the closure of that open orbit is the irreduc-ible component. However, for the even dimensional modules thereis an additional irreducible component not given by an open orbit.It is the one containing the orbits of the modules of the form

where Σi5 deg {πj) = m. In general — when the polynomials πlf , πr

are all distinct — the codimension of the orbit is m.In [5] Flanigan and Donald give a formula for the number of

irreducible components in Mod (̂fcm). The number is

1 for m = 1

2[(m + 3)/6] for m > 1 and odd

2[m/6] + 1 for m even

where the square brackets denote the greatest integer function.

6. Deformation theory of modules* The theory of deforma-tions of modules follows in the spirit of Gerstenhaber's work ondeformations of algebras in [8]. Viewed in a geometric settingdeformation theory concentrates on the &[[£]]-valued points of thescheme Mod^(M). These points are the deformations, and we willcall them formal deformations. (In [4] the term used is "genericdeformation.") Strictly speaking the functor Mod̂ (-M) need not even

214 KENT MORRISON

be representable; for example, M could be infinite dimensional, andthe ideas of deformation theory could be applied while the algebraicgeometry could not. However, no work has been done in situationswhere M.oάA(M) is not a scheme, even though that fact is not used.

Suppose we are given an actual curve or one-parameter family ofmodule structures, that is, a scheme morphism c: Specft[£]-*Mod4(ikf).Taking the Taylor expansion of c at 0 gives a formal deformationof the module structure p — c(0). Functorially, we are consideringthe localization morphism a: k[t] -» ftp] inducing Spec a: Spec ftp] —>Specftp] and the composition c°Specα, which gives a ftp]-valuedpoint of ModA(M). Then we define the set of formal deformationsof p to be the scheme morphisms β: Spec ftp] —> MoάA(M) such thatβ(x) — p where x is the unique closed point of Spec ftp]. Alge-braically β is simply an A (x) ftp]-module structure on ikf(x)ftp]which restricts to p when t = 0. Let π: ftp] —> ft: 11-> 0 be the pro-jection. Then the formal deformations of p are Mod^(M)(ττ)~1(^).Not every formal deformation need be the Taylor expansion of anactual deformation; over the complex field there are convergenceproblems.

It is customary (see [4]) to define a formal deformation of pto be an A (x) if-module structure on M(g)K, K = ft((ί)), of theform

pt{a){m) — p(a)(m) + tR^aXm) = fR2(a)(m) + -. •

for aeA, meM, Rt: A —> Endfc(M). Then it is extended to A®Kand M(g)K to make it i£-linear in each factor. The advantage inthis definition is to allow everything to be done over a new basefield K. However, it is not geometric because there is no projec-tion K—>k, so there is no way to recover p from an arbitrary A(§)if-module structure on M (x) K. The two different definitions areequivalent: from a ftp]-point we get a module structure on M(x)Ksimply by extending the scalars. In the other direction we needto know that pt is of the special form above, which says that pt

is actually defined over ftp].In a similar fashion one can define the formal deformations of

any ft-valued point x in a ft-scheme X as the set of ftp]-pointslying over x. For example, the set of formal deformations of theidentity I in Antk(M) will be of interest to us. This set forms asubgroup of Autfc(Λf)(ftp]) and there is an action of the group ofdeformations of / on the set of deformations of p. Explicitly, adeformation of I can be written in the form I + tut + fu2 +where each ut is in EndΛ(Λf). As with deformations of p we maydefine a formal deformation of / to be a if-linear automorphism ofM(x)K having the form above.


One says that two formal deformations pt and σt are equivalentif they are in the same orbit for the action of the group of formaldeformations of /. A module p is rigid if every formal deforma-tion of p is equivalent to the zero-deformation (the canonical exten-sion of p to M®k\tJ). That means that all formal deformationsof p are equivalent, so there is just one orbit in the set of formaldeformations of p.

Immediately one sees that if the orbit of p is an open sub-scheme of Mod^ikf), then p is rigid. At p the &[£]-points of Xandthe kit]-points of Mod^(M) are the same, kft] being a local algebra.

If p is a nonsingular point of Mod (̂Λf) then the converse isalso true. Let v be a tangent vector at p. Then v is tangent toan actual deformation, which is a curve through p, and thereforethere is a formal deformation pt — p + tv + . . . . But pt is equivalentto the zero-deformation by an automorphism / + tuL + . An easycalculation shows that the cocycle v is the coboundary of u19 i.e.,v{a) = [ρ(a\ u,]. Therefore H\A, End/C(M), p) = 0 and it follows thatXp is open.

If p is a singular point it may be that p is not reduced, soconsider the reduced subscheme. If p is nonsingular in the reducedscheme and if p is a rigid module, then the orbit Xp is open inModΛ(M)ΐeά by a similar argument.

Over an algebraically closed base field k rigidity of the modulestructure p is equivalent to the orbit Xp being open. If k is notalgebraically closed one should consider the i?-points for all completediscrete valuation rings R.

PROPOSITION 6.1. Let k be algebraically closed. The modulestructure p is rigid if and only if Xp is open in the reduced sub-scheme

Proof. Let X = XP and Y = MoάΛ(M)rea and x = p. We havethe situation in which I c Γ is a subvariety and x e l is a pointat which the A;[ί]-points of X and Y are the same. If X is notopen at x, then X is a proper subvariety in a neighborhood of x.Let Z be any subvariety of Y such that x is isolated in Xf]Z.Then there is a nontrivial morphism a: Spec k\t\ —> Z such that a(p)~x where p is the closed point of Specfc[ί]. Such an a is a k\t]-point of Y based at x which is not a k[t}-point of X. •

The following is an example of a module structure which isrigid and whose orbit is open in the reduced scheme but not inMod^(M). Let A be k[Zp] where A: is a field of characteristic p, andlet p be the trivial one-dimensional representation. Then

216 KENT MORRISON

Spec k[x]/(xp — 1) which has one closed point, the representation p,but is not reduced. The orbit of p is the closed point, the sub-scheme corresponding to Spec k[x]/(x — 1), but it is not open becauseit does not fill up the tangent directions. In this case the orbit ofp and the reduced scheme are the same.

Several of the results proved in [4] about deformations ofmodules can be proved in the context of the scheme of modulestructures. We will give several examples in which a property ofmodules is stable under formal deformations because it is actuallyan open condition, a property which is true on an open subschemeof ModΛ(M). Such open properties are cyclic, faithful, irreducible,protective, and injective.

First, consider a cyclic A-module structure p with generatorxeM. The evaluation map evx: A —> M: α π p(a)(x) is surjective. Onewould expect that x is a generator for any module structure insome open neighborhood of p. Define the scheme morphism

ev: Mod (̂Λf) X M > Hom*(A, M)

where ev(p, x) is the morphism that sends a to p(a){x). The functorHomfc(A, M) is represented by the affine λ -scheme Spec £(A* ® M)where S denotes the symmetric algebra. This scheme is of finitetype if and only if A is finite dimensional. Now in Homfc(A, M) isthe open subscheme of surjective maps Surfc(A, M) whose iϋ-pointsare the surjective 22-module homomorphisms from A (R) 22 to Λf®22.Thus βv~1(Surfc(A, Λf)) is an open subscheme of Mod^(M) x M con-sisting of pairs (p, x) for which x is a generator for p. Also, wemay fix the generator x. Then evîSurÂ, M)) is open in thescheme of module structures and contains p; this set consists ofthe modules that are cyclic with # as a generator.

Not surprisingly, for formal deformations pt of p, the elementx (x) 1 is a generator for pt. This is obvious from the algebraicgeometry.

Recall that a module is faithful if its representation map p:A —> Enάk(M) is injective. Since M is finite dimensional, A mustalso have finite dimension. In order to show that the faithfulmodules form an open subscheme of ModA(M) consider the opensubscheme SInjfc(A, M) contained in Homfc(A, M) whose ϋ?-points arethe split injective maps φ: A(g) R-+ M(g) R. Clearly for 22 = k weget the injective maps from A to M. To see that SlnjÂ, M) isreally a scheme, we identify it with SurΛ(ikf *, A*) where Λf * andA* are the Jfc-duals of M and A. A morphism φ: A (g) R —> jfcf® Ris split injective if and only if *φ\ (Λf® 22)* —> (A (g) 22)* is surjec-tive. If we only required that the 22-points be injective maps then


the functor would not be representable.Now the subscheme SInjA.(A, M) Π Mod^(M) is open in MoάA(M)

and the fe-points are the faithful A-module structures on M. Theϋ?-points are the split faithful A (x) ϋ?-module structures on M®R.It is immediate that any formal deformation of a faithful moduleis faithful.

We will briefly consider the classes of simple modules, projec-tive modules, and injective modules. Let k be algebraically closedand let p: A —> Endfc(ikf) be a simple A-module structure. Sinceevery closed point in the connected component of p has the sameJordan-Holder factors as p, we see that the orbit of p is actuallythe connected component containing p in the reduced subschemeModA(M)ΐed. We also see that p is rigid.

It is unclear how to use this argument to show that simplemodules are rigid when k is not algebraically closed. There is,however, an algebraic proof in [4].

Now since ΈxtA(E, F) = 0 whenever E is injective or F isprotective, the orbits of injective modules and the orbits of pro-jective modules are open, and therefore injectives and projectivesare rigid.

REFERENCES

1. M. Artin, On Azumaya algebras and finite dimensional representations of rings, J.Algebra, 11 (1969), 532-563.2. C. I. Byrnes and M. A. Gauger, Decidability criteria for the similarity problem, withapplications to the moduli of linear dynamical systems, Adv. Math., 25 (1977), 59-90.3. M. Demazure and P. Gabriel, Groupes algebriques, Tome I, North Holland, Amster-dam, 1970.4. J. Donald and F. J. Flanigan, Deformations of algebra modules, J. Algebra, 31(1974), 245-256.5. J. Donald and F. J. Flanigan, The geometry of Rep (A, V) for a square-zero algebra,preliminary report, Notices Amer. Math. Soc, 24 (1977), A-416.6. P. Gabriel, Finite representation type is open, Proceedings of the International Confer-ence on Representations of Algebras, vol. 488, Lecture Notes in Mathematics, Springer -Verlag, New York, N. Y., 1976.7. I. M. GeΓfand and V. A. Ponomarev, Remarks on the classification of a pair of com-muting linear transformations in a finite-dimensional space, Functional Anal. Appl., 3(1969), 325-326.8. M. Gerstenhaber, Deformations of rings and algebras, Ann. Math., 79 (1964), 51-103.9. A. Heller and I. Reiner, Indecomposable representations, Illinois J. Math., 5 (1961),314-323.10. P. J. Hilton and U. Stammbach, A Course in Homological Algebra, Springer-Verlag,New York, N. Y., 1971.11. K. Morrison, Schemes of module structures and deformations of modules, Ph. D. Dis-sertation, University of California, Santa Cruz, June, 1977.12. D. Mumford and K. Suominen, Introduction to the theory of moduli, Proceedings ofFifth Nordic Summer School in Mathematics, Oslo, 1970, ed. F. Oort, Wolters-Noordhoff,Groningen, Netherlands.

218 KENT MORRISON

13. A. Nijenhuis and R. W. Richardson, Jr., Deformations of homomorphisms of Liegroups and Lie algebras, Bull. Amer. Math. Soc, 73 (1967), 175-179.14. C. Procesΐ, Finite dimensional representations of algebras, Israel J. Math., 19 (1974),169-182.

Received November 1, 1978.

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LOGAN, UT 84322

Current address: California Polytechnic State UniversitySan Luis Obispo, CA 93407

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